Tag Archives: fractals

(This article originally appeared in Get Ahead Magazine, for the Get Ahead Festival of independent short films in Brooklyn.)

When we speak of the identity of a place, we express a recognition of the patterns formed around us. We may not be conscious of them to the point of being able to draw them back with precision like Stephen Wiltshire, but we can remember them in the abstract, and in this way, identify different places from the abstractions we recall of their patterns. This is how one street can look sufficiently alike another that we can identify a neighborhood, and it is also why a landscape like Liberty City in Grand Theft Auto can feel like New York City, despite the fact that every object has been reconfigured to create a parody environment.

A city’s identity is made by the patterns selected by the people who built them. We can also say that these patterns are the fossil record of the people who inhabited a place. We can read the history, the culture and the sustainability of a place by the combination of its patterns. A building is a hierarchical computation of different processes nested within each other, and these processes can be substituted for others depending on what conditions are encountered.

Echoes of Holland

At the largest scale of patterns there is the building program, whether a house, a church, an office, easily recognizable in any cultural setting. These programs are realized using construction techniques that are conditioned on economic constraints. The Dutch who settled New York City brought with them their basic house program, but these had to adapt to the resources available by, for example, building in brownstone, an economic pattern. Despite this difference they kept features of their homes like stoops, patterns that were at first environmental but then became cultural.

As each successive culture either migrates to or emerges in the city, it needs to adapt the patterns of its buildings to fit its own practices. Fractals like these become habitual:

This is Chinatown in Brooklyn. We can tell it is Brooklyn because the basic patterns, program and materials, are Americanized Dutch. We can tell it is Chinatown because of the use of vertical commercial signs which are characteristic of oriental cultures (their writing being read top-to-bottom instead of left-to-right). The large-scale patterns are extended by smaller-scale patterns to form a full building fractal that is Dutch, American, New York and Chinese. This combination of pattern is the identity of Brooklyn, the people who have lived there and continue to live there.

One particular culture that has often been denounced as an anti-culture is the global corporation. Their aesthetic program has been to impose their corporate identity uniformly on communities, regardless of any consideration for local economic, environmental, or cultural factors. But there have been exceptions, such as the following case, where the corporation decided to extend the patterns of the neighborhood instead of imposing its own.

This Dunkin’ Donuts nested itself seamlessly in an old Dutch building next to a Chinese restaurant, and even improved upon it a bit with orange awnings that preserve the structure of the windows while announcing the presence of this corporate neighbor to everyone on the street. As well as being a demonstration of Dunkin’s neighborliness, it is also a demonstration of the sustainability of the neighborhood. The buildings are resilient, and despite the Dutch builders never anticipating that there could ever exist such a thing as a Dunkin’ Donuts, their patterns have been slightly adapted to fit today’s needs. Some day Dunkin’ Donuts will also be history. In its place will be some other culture which may or may not preserve traces of Dunkin’s presence, but the building itself will remain and serve a new purpose.

So far I haven’t said a word about architecture, which is simply because architecture does not enter the picture unless one has a lot of money for sculptural elements. It is possible to build a good neighborhood without architects, but a great one needs art, and that means getting some architects involved. The best architecture starts with utilitarian patterns, the same functional, economic and cultural patterns we see above, and then expands it by nesting sculptural elements, thus it is still possible to recognize identity of place behind the architecture. This architecture, sculpting the utilitarian shape of the building, becomes the final expression of identity, the artistic currents and fashions that propagate across cultures and then vanish, only to make periodic comebacks.

This is what Brooklyn architects did with these residential towers overlooking prospect park. What is in essence a stack of identical apartments made with the usual economic patterns was extended with sculptural ornament, most impressively around the otherwise obnoxious elevator shafts.

Looking at Brooklyn’s tallest landmark, the Williamsburg Savings Bank Tower, we see patterns that are Gothic, Romanesque, Italian renaissance, Art Deco, all nested within each other and wrapped around a stack of floors that can fulfill any purpose whatsoever. The final product is a building that is worth preserving from a bank, to dentist’s offices, to residences, because the patterns cooperate with each other instead of clashing, and answer our need to feel connected to any of these identities. This is another form of sustainability.

The tragedy of architecture in the 20th century, and the great confusion that came from it, is that modernist architects first banned sculptural elements in favor of purely standardized, globally uniform, utilitarian industrial patterns, then post-modernist architects declared that a building was only a sculpture for living, that the utilitarian should be subordinated to the architect’s artistic expression. The outcome has been a building culture that has no identity when it is not completely incomprehensible, and more than likely has no resilience and no future.

This “Dance Center” is a sad example of this confusion. Were it not spelled out in letters, would we be able to understand anything about this building’s identity? The people who occupy it? What any of its parts do, or if they do nothing at all and are simply there for visual effect? I can’t imagine a future for it. But there is worse.

This building makes no attempt at being anything other than mass human storage, the modernist tower block revived for the bubble epoch. It will likely be a financial failure for being too ambitious while being too redundant. If I were to take an apartment there, it would be impossible for me to tell which window is mine from the outside. What does this say about the people who built it? That they took the easiest path to financial income. What does this say about the people who will live there? That they have nowhere else to go. It is and will remain an alien in the neighborhood, a product that removes identity instead of contributing to it.

Today’s planning establishment attempts to reform the shape of our cities with “form-based codes” that dictate with precision the shape of every pattern. This comes at the cost of outlawing certain unforeseeable patterns that may make a net contribution to the identity of place. It also drives away people that need these patterns, and drains life that is needed to renew the neighborhoods. Last of all, it will not stop a monster like the example above. If instead of dictating shapes, we made it clear how to expand and preserve the neighborhood’s identity, we would all be much freer to live and express ourselves, adding to the history of our environment.

I don’t believe that there is a dichotomy between a supposedly modern and traditional architecture. Instead there exist different geometric processes, and while traditionally builders have employed nesting processes in their work, for perhaps no other reason than it came naturally to them, modern builders have restricted themselves to linear geometric processes due to drawing their inspiration from Cartesian science and engineering.

In attempting to transform architecture into a vessel for artistic expression, modern architects have been trapped by their limited tool set, and the product of their work has often been confusing, silly, or utterly corrupt. There are only so many tricks that one can perform with linear geometry, although computers have extended the reach of those tricks. But the confusion of modern architects becomes even more obvious when they ascribe artistic merits to traditional builders who never aspired to be artists at all. One such instance is the introduction of a recent biography of the 18th century french military engineer Vauban by official starchitect Jean Nouvel, who described Vauban’s fortresses as an early form of land-art and morphing. Jean Nouvel asks, could a man be an artist without being aware of it? Vauban was not an artist at all. Military necessity led him to employ geometric processes that significantly increased the complexity of fortifications, and it is merely incidental that today we find his projects to have artistic merits.

The process through which Vauban’s work became worthy of architectural praise provides the key to the distinction between linear and nesting geometry. Vauban was not himself the inventor of the star fort. Those had been around for more than a century when he began his career for the army of king Louis XIV. The basic star fort was a simple concept: the old masonry walls of the medieval age had shown themselves to be obsolete with the advent of cannons, and they had been replaced with thick banks of earth dug out of trenches whose major flaw was to provide space out of reach of defensive fire at its angles. The angles were thus extended into diamond-shaped turrets in the first pass at a feedback correction, introducing nesting geometry and initiating the first step of the genesis of a fractal.

A basic, early star fort

While the star fort was successful at resisting attacks, it was not impregnable. A method was devised to capture them by digging trenches in zig-zagging patterns through which troops could assault the walls without being exposed to cannon fire. In fact this is how Vauban built his career, and some of his “plans” for besieging star forts are significant civil engineering projects of their own.

The siege of Turin. From an encircling trench, Vauban built successively denser trenches to capture the citadel and take the city, a process that was extremely expensive and time-consuming.

While star forts never truly became obsolete (as medieval fortifications had) until well into the 19th century, military engineers did improve on their effectiveness by correcting their vulnerabilities, which happened to be at the angles they were characterized by. And so, by another layer of feedback, the geometric depth of the star fort concept increased.

San Martin Citadel, a “second generation” star fort.

Vauban’s great invention was nothing much more than repeating this process of increasing depth one more time, creating what many now consider to be his masterpiece, the Citadel of Lille, a showcase of complex geometry made from the refinement produced by centuries of feedback of the star fort concept.

Citadel of Lille and the system of fortification of the City of Lille, as designed by Vauban

If you only understand Cartesian processes, then the only idea that may come to you to improve on the basic star fort would be to add dozens of diamond-shaped turrets, a change that would most certainly make the concept worse instead of better. The military engineers of the time however were well aware that the diamond turrets were optimal in their shape. What was needed was a shape that extended the diamond, and this was achieved by increasing the depth of the whole object.

Another aspect of the complexity of a geometric process seen in the Lille example is its configuration adaptiveness. The shape of the city and the surrounding landscape is completely random, and the encircling fortifications bend to match this randomness, leading to Nouvel’s claim that it is an early example of morphing. But once again there is no deliberate attempt at morphing going on. Since each component of a star fort is defined as a recursive relational transformation of the basic wall, Vauban only had to design the wall and the other parts aligned themselves as a result of the wall’s configuration. If the outcome has artistic value, it is once again only incidental.

It is important to note that the Vauban extensions to star fortifications did not mean that the simple 3-part star fort became obsolete. In fact many simple star forts were built in the 18th and 19th century in America as the threat was low and the cities to be defended underdeveloped. The difference between a simple fort and Vauban’s complex fort is one of depth and effectiveness, and there is a real cost-benefit choice to make. The star fort only became obsolete when the bunker replaced it, and the early bunkers reset the process of complex geometry genesis by being simple concrete shells in their early incarnations.

When we undertake to create symmetry in an urban environment, we want buildings to be as alike as possible while allowing for adaptation to context. If we understand geometric depth we can build in such a way that poor and expensive buildings have the same basic design in their first levels of geometry, but expensive buildings have many more scales of geometry nested within that basic design. It is not necessary for an entire city to be made of the same materials as materials are one of the last visible scales of geometry, and so we can have a city of mud bricks and marble buildings that nevertheless share 95% of their geometry and beautifully complement each other, while both poor and rich citizens have a home adapted to their situation.

We can look at these examples from Korean traditional architecture for an illustration.

On the left is a simple house and on the right is the tomb of a great king. Both buildings have the same design, but the building on the right has much greater depth in this design.

Another interesting comparison is between the Golden Gate bridge in San Francisco and the Verrazano Narrows bridge in New York.

The bridges are the same in design, but the Golden Gate bridge has more depth within this design, and is for this reason the more famous of the two bridges. That doesn’t mean the Verrazano Narrows bridge isn’t beautiful on its own.

And to make things as simple as they can get, we can compare a Sierpinski triangle with four levels of iteration with one that has six levels.

The fractal on the right has all the same elements as the one on the left, but also has more.

A lot of the residential buildings we create today would benefit from being more like the Verrazano Narrows bridge. They try to be more than a simple house for a simple family and end up covered in tacky, useless ornament that have obviously been forced into the design. Simplicity, if it is adapted to context, can create as beautiful a landscape as complexity. Postmodernistic nonsense geometry does not. We would be better served going back to the simplicity of 1950’s international style modernism than what is being built by architects today. The best architects would reinvent it with greater depth.

The following story about a presidential program to demolish whole neighborhoods of inner city fabric in the United States and turn them back into wilderness has been making the rounds around news blogs.

Having outlined his strategy to Barack Obama during the election campaign, Mr Kildee has now been approached by the US government and a group of charities who want him to apply what he has learnt to the rest of the country.

Mr Kildee said he will concentrate on 50 cities, identified in a recent study by the Brookings Institution, an influential Washington think-tank, as potentially needing to shrink substantially to cope with their declining fortunes.

Most are former industrial cities in the “rust belt” of America’s Mid-West and North East. They include Detroit, Philadelphia, Pittsburgh, Baltimore and Memphis.

In Detroit, shattered by the woes of the US car industry, there are already plans to split it into a collection of small urban centres separated from each other by countryside.

“The real question is not whether these cities shrink – we’re all shrinking – but whether we let it happen in a destructive or sustainable way,” said Mr Kildee. “Decline is a fact of life in Flint. Resisting it is like resisting gravity.”

This is the type of neighborhood that the government wants to disurbanize. It is located in central Detroit.

To someone trapped in the mindset of development and control that we have practiced in the 20th century, a place like this is a nightmare. It is not possible to consolidate properties in order to bring in a large developer and a large bank that will finance “re-development” of the place. Worse yet, properties have been abandoned randomly, turning what were neat row houses with identical lots into a pockmarked landscape of randomly-sized public land chaos. Better to demolish everything and start over.

There is another mindset through which to interpret such a neighborhood, that of complexity. If we embrace complexity, then the randomly sized pockets of open land are an exceptional opportunity to renew the city of Detroit. They form a fractal solution set to new construction that many different people can participate in and contribute to. It can accomodate small, medium-size and eventually large-size businesses in close proximity with diverse housing and convenient transportation structures.

But why has this not worked for Detroit? Because its process of growth has not been focused on fractal scales but only on big projects and big businesses. Now that the big businesses are dying the city is threatened with disappearing and has to beg even bigger governments to prevent their death. That cannot go on forever. Death is a normal, natural process, and big businesses disappearing should never be a threat to a large city. The economic fabric of a city must always be renewed by new businesses. It is this renewal that creates a sustainable business ecology. At some point Detroit stopped the process of new business creation, and from then on its decline was assured.

Instead of demolishing its remaining neighborhoods and surrendering to the decline and death that will surely follow in its reduced form, Detroit should instead adopt the process of a special economic zone in those neighborhoods it wants to return to “nature”. Tolerate people build as they wish and let a slum happen, and from the slum will emerge the businesses that will renew Detroit’s economy. It can’t be worse than the bulldozer.

1. Sun Tzu said: The control of a large force
is the same principle as the control of a few men:
it is merely a question of dividing up their numbers.

2. Fighting with a large army under your command
is nowise different from fighting with a small one:
it is merely a question of instituting signs and signals.
– From The Art of War by Sun Tzu

The problem of social cooperation is how to order many individuals into large-scale patterns, and thus acquire the benefits of these larger patterns. The military arts were the first to face this problem, war being a field where inferiority carries severe consequences, and lessons are learned quickly. The solution was known in the time of Sun Tzu: the superior army was the one that could act as a single force, applying a single decision multiplied by however many men were at the command of this army. More men were always better, but past a certain scale it became unmanageable for a commander to yell out orders to everyone and maintain command. In order to resolve this the military men invented hierarchy, a command structure through which the commander’s orders would be distributed so that a group of any size could act as a single force.

For most of history success in war came from achieving and maintaining organization, lines of command from a center to the individuals that compose an army such that the commander could deploy the army in the most effective pattern he could think of. Discipline and complete obedience to orders was required, even if the situation as it appeared to the lowly grunt was in total contradiction to the orders he had been signaled. As far as he knew, the commander had a larger picture of the war and the orders ought to work out correctly. But the flaw in organization is that as an organization becomes larger, as the layers of hierarchy increase, the commander becomes more remote and more isolated from his army. The lines of communication become inefficient, the orders become irrelevant, and many men die stupidly.

Nevertheless, for centuries the sheer overwhelming force of numbers more than made up for the losses due to bad orders. The principle of organization triumphed. Reformers started looking for plans to organize industries, entire nations (the command economy of the Soviet Union), and of course, cities. The C.I.A.M. Athens Conference resulted in the publication in 1942, by Le Corbusier, of the Athens Charter, the document upon which the plans to organize modern cities, and be rid of the spontaneous historic city, were founded.

Between the time of the Athens Conference and the publication of the Athens Charter, the military concept of large-scale organization was completely discredited.

In June 1940 the German army invaded France. The two armies were evenly matched in men and weapons, France even having a advantage in tanks. Within one month the French army organization collapsed and millions of men surrendered without having put up much of a fight, resulting in many decades of American jokes about French surrender. In reality the two armies were far from evenly matched; the German generals had discovered a mean to overcome the weakness in the principle of organization, that it relied on a central, single commander. Their model of cooperation has been called Blitzkrieg, the lightning war, and its intent was to reduce the delay in receiving and sending the “signs and signals” of command by removing them. German commanders out in the field were given broad directives and trusted to figure out on their own how to fulfill them, with glory and medals as reward for success. The French had instead refined organization and bureaucracy into a precise art. Within days of breaching into France, autonomous German tank divisions destroyed the lines of communication of the French army and paralyzed the front-line units. It became impossible for it to act as a single force, never mind stopping an invasion.

The German system of directive command was in fact the universal principle of emergence applied to military action. Instead of building a hierarchy of orders to communicate the will of a central commander, the armies were organized in parallel, directed to respond to their observed context, a context which was itself produced by other units of the same army. Instead of deploying the intelligence of a single commander holed up in an office in Berlin, the German system linked the intelligence of all of its officers into a more effective super-intelligence that could see all of the battlefield simultaneously. The collapse of the French army was therefore inevitable. It was a case of one against many.

As already mentioned, war teaches quickly, and the allies eventually adopted a similar operations model to fight the war to victory. German operations theorists went on to design the structure of NATO’s European defense, a war that we fortunately never witnessed. Urban planners did not have to learn this lesson, and they opted to organize cities to ruin.

The network structure is often, incorrectly, called a “bottom-up” organization. My opinion is that this label makes no sense. There is no up or down in a network. There is neither bottom nor top. Those are descriptions that apply to hierarchies only. In a network actions happen horizontally, in parallel. Large-scale patterns are made up of links between those local actions, as seen in the figure above. Human intelligence, for example, cannot be explained as a collection of cells. It is the patterns formed by the links between these cells that is intelligent, and it is these patterns that allow us humans to be several orders of magnitude more complex than individual cells.

The paralysis inflicted on the French army organization was in parts self-inflicted. Longer chains of command involved delays in transmitting information (reports from the field), analyzing the information, planning a reaction and ordering the new deployment. The bigger the army became, the more paralysis it suffered. This organization was in much the same situation as the dinosaur who did not feel a hit on his tail because the nerves were too far from his brain. The bigger it became, the more exposed it was to a paralysis-focused attack.

It should not come as a surprise that what caused the death of cities is also self-inflicted paralysis. But the case of cities is much more tragic. The German operations model was novel and innovative, a radical improvement in military art. Cities, however, had always been emergent. They were the product of a spontaneous order, a phenomenon that was barely understood at the height of rationalist planning. What science did understand was organization. Since it was accepted as the pinnacle of science, no rational thinker could reject the new urban planning. The planners did not notice the hints: what they were organizing had not been a creation of anyone.

In a complex emergent system, the number of unique patterns scales up with the size of the system. (What some emergence commentators call “more is different,” another expression that makes no sense.) While an organization attempts to create a large-scale pattern to outmatch smaller patterns, a complex system is made up of both small and large patterns, in proportion to a power law, either nested together or juxtaposed randomly (a fractal). If an emergent system is intelligent, it will structure itself into patterns that no one had expected.

For centuries people had been accustomed to such patterns as the street of similar shopkeepers. Many streets in European cities bear the name of a particular trade, such as baker’s street or threadneedle street. But when cities passed a critical scale during the industrial revolution, a whole new pattern emerged: the central business district. An entire city within the city became the center of commerce, not simply specific streets next to residences. Although it appeared unexpectedly during the 19th century (the Haussmannian renovation of the Opera district of Paris was meant to create a neighborhood for the upper classes, but it became a business center immediately and has remained so ever since), a central business district came to be what a major city was all about. When planners set out to organize a modern city, they planned it around the CBD as the central feature. They did this by drawing a square on the map and applying a different set of rules to this square. Within a few years, their CBDs began dying. The small scale patterns nested within them had been zoned out.

In retrospect it was inevitable for an attempt at organization to severely interfere with urban processes, the principle of organization being a step down in complexity from the principle of emergence. Organization had a sinister advantage: it gave the planners the illusion that they could predict what the city was going to become. An emergent system cannot be predicted with precision. The very basis of its intelligence is that it has not yet been decided what it is going to do. Embracing an emergent system means accepting that patterns will appear that are beyond our comprehension. (In Wolfram’s terminology, the system is computationally equivalent to our own intelligence.)

By trusting their front line officers to run the war for themselves, the German general staff took a leap of faith that paid off decisively and confronted every opposing military with their crippling inferiority. I suspect the first modern city to give up on the principle of organization will trigger a similar revolution.

The qualities of an emergent city

The adoption of mass-production processes, or development, in substitution for spontaneous urban growth in the mid-20th century created for the first time a phenomenon of alienation between the inhabitants and their environment. While the physical features of spontaneous cities could be traced to complex histories of families, businesses, and organizations, the physical features of planned cities owe their origin only to the act of planning and speculation. This has severe consequences towards the sustainability of place as there will not grow any particular attachment by the residents, their presence there being only a temporary economic necessity and not the outcome of their life’s growth. Mass-production of the environment left people as nothing more than consumers of cities where they used to be their creators. A building culture was replaced with a development industry, leaving the landscape culture-less and with no particular sense of identity. This took place despite the evidence that a building which has a unique history and has been fitted to someone’s life, as opposed to speculatively produced, generates market value for that property. (Alexander, 1975) This is why, although the demolition of so-called “slums” to replace them with modern housing projects created a great deal of opposition against urban renewal programs, the demolition of the housing projects later on did not lead to a popular preservationist opposition. They were not the physical expression of any culture.

In additional to cultural patterns, spontaneous settlements also have a peculiar morphology that has not successfully been imitated by modern growth processes. Spontaneous settlement processes give individuals full freedom to determine the boundaries of their properties. Spontaneous settlement is one where total randomness in building configuration is allowed, with no pre-determined property lines acting as artificial boundaries. Buildings and building lots as such acquire general configurations comparable to cell structure in living tissues, unique sizes and boundaries that are purely adapted to the context in which they were defined. In the absence of abstract property boundaries, property rights are bounded by real physical limits such as a neighbor’s wall. (Hakim, 2007)

Very attractive spontaneous cities have a specific pattern of the urban tissue. It consists of similar vernacular buildings that appear very simple when considered individually, but produce a visually fascinating landscape when considered as a whole. This is a form of fractal geometry. In mathematics a fractal is a geometric object of infinite scale that is defined recursively, as an equation or computation that feeds back on itself. For example the Sierpinski triangle is defined by three triangles taking the place of one triangle as in figure 4.

Figure 4. A triangle triggers a feedback function that produces three triangles, which themselves trigger the feedback function to produce nine triangles, and so on. This process can unfold as long as computational resources can be invested to increase the complexity of the object.

The Mandelbrot Set is a much more interesting fractal that is defined as a simple recursive mathematical equation, yet requires a computation to visualize in its full complexity. When computing how many cycles of feedback it takes for the equation to escape to infinity for specific coordinates, figure 5 is the outcome.

Figure 5. The image on the right is a deeper magnification of the image on the left, produced with a narrower range of coordinates as the input of the Mandelbrot set’s feedback function.

In addition to its remarkable similarity to natural phenomena, this form of geometric order informs us of a very important law in geometry: a feedback loop that is fed through the same function will produce an ordered but unpredictable geometric pattern out of any random input.

This tells us why cities of vernacular buildings have such appealing geometric properties at the large scale, despite being often shabby and improvised at the scale of individual buildings. Shanties made of scrap metal and tarp look rough at the scale of the material, but because multiple shanties share the construction process and originate from similar feedback conditions they form an ordered geometric pattern with its specific “texture”. The same process takes place at other scales of feedback, for example the production of a door. Whether the input for one door is larger, taller, wider than another door, if the same production process is employed the two doors will contribute to the overall fractal order of the urban space. This law has been employed not only in traditional and spontaneous cities, but also for modern urban planning initiatives. In the New York City neighborhood of Times Square the structure of billboard advertisements is defined by a building code that determines their configuration in relation to the configuration of the building. The outcome is a unique tissue of advertisement billboards that has become more characteristic of the neighborhood than the buildings themselves, which are not produced by a shared feedback function.

Fundamentals of urban complexity

Christopher Alexander showed in A City is not a Tree (Alexander, 1965) that social and economic networks formed complex semi-lattice patterns, but that people who observed them limited their descriptions to a simple mathematical tree of segregated parts and sub-parts, eliminating connections in the process. (Figure 6 compares the structure of a tree and semi-lattice.) In attempting to plan for urban structure, a single human mind, without a supporting computational process, falls back on tree structures to maintain conceptual control of the plan, thus computing below spontaneous urban complexity, a phenomenon that is consistent with Wolfram’s theory of computational irreducibility of complex systems. (Computational irreducibility states that the only accurate description of a complex system is the system itself and that no abstraction or reduction to a simpler process is possible.) Nikos A. Salingaros later detailed the laws of urban networks in Theory of the Urban Web. (Salingaros, 1998) Network connections form between nodes that are complementary, and therefore the complexity of networks depends on an increasing diversity of nodes. Salingaros describes the urban web as a system that is perpetually moving and growing, and in order to do this the urban tissue has to grow and move with it. Consider for example the smallest social network, the family. Debate over accessory units or “granny flats” has intensified as normal aging has forced the elderly out of their neighborhoods and into retirement complexes, while at the other end of the network young adults entering higher education or the labor market vanish from a subdivision, leaving a large homogeneous group of empty-nesters occupying what was once an area full of children, and often forcing school closures (a clear expression of unsustainability).

Figure 6. A comparison of a tree pattern on the left and a semi-lattice pattern on the right. The tree structure is made of groups and sub-groups that can be manipulated separately from others. The semi-lattice pattern is purely random without distinct sub-parts.

These social networks grow more complex with increasing building density, but a forced increased in density does not force social networks to grow more complex. For instance the spontaneous settlements of slums in the developing world show remarkable resilience that authorities have had difficulty acknowledging. Because of squalid living conditions authorities have conducted campaigns to trade property in the slum for modern apartments with adequate sanitary conditions. To the authorities’ befuddlement some of the residents later returned to live in the slum in order to once again enjoy the rich social networks that had not factored in the design of the modern apartments and neighborhoods, demonstrating that the modern neighborhoods were less socially sustainable than the slums.

In commercial networks, space syntax research (Hillier, 1996), using a method for ranking nodes of semi-lattice networks, has shown that shops spontaneously organize around the multiple scales of centrality of the urban grid at its whole, creating not only commercial centers but a hierarchy of commercial centers that starts with sporadic local shops along neighborhood centers and goes all the way to a central business district located in the global center of the spatial network. The distribution of shops is therefore a probabilistic function of centrality in the urban grid. Because the information necessary to know one’s place in the hierarchy of large urban grids exceeds what is available at the design stage, and because any act of extension or transformation of the grid changes the optimal paths between any two random points of the city, it is only possible to create a distribution of use through a feedback process that begins with the grid’s real traffic and unfolds in time.

The built equilibrium

Although they may appear to be random, new buildings and developments do not arise randomly. They are programmed when the individuals who inhabit a particular place determine that the current building set no longer provides an acceptable solution to environmental conditions, some resulting from external events but some being the outcome of the process of urban growth itself. It is these contextual conditions that fluctuate randomly and throw the equilibrium of the building set out of balance. In order to restore this equilibrium there will be movement of the urban tissue by the addition or subtraction of a building or other structure. In this way an urban tissue is a system that fluctuates chaotically, but it does so in response to random events in order to restore its equilibrium.

This explains why spontaneous cities achieve a natural, “organic” morphology that art historians have had so much difficulty to describe. Every step in the movement of a spontaneous city is a local adaptation in space and time that is proportional to the length of the feedback loops and the scale of the disequilibrium. For spontaneous cities in societies that experience little change the feedback loops are short and the scale of disequilibrium small, and so the urban tissue will grow by adding sometimes as little as one room at a time to a building. Societies experiencing rapid change will produce very large additions to the urban tissue. For example, the skyscraper index correlates the construction of very tall buildings with economic boom-times, and their completion with economic busts. The physical presence of a skyscraper is thus the representation of a major disequilibrium that had to be resolved. (Thornton, 2005) The morphology of this change is fractal in a similar way that the movement of a stock market is, a pattern that Mandelbrot has studied. In general we can describe the property of a city to adapt to change as a form of time-complexity, where the problems to be solved by the system at one point in time are different from those to be solved at a later point in time. The shorter the time-span between urban tissue transformations, meaning the shorter the feedback loops of urban growth, the closer to equilibrium the urban tissue will be at any particular point in time.

Modern urban plans do not include a dimension of time, and so cannot enable the creation of new networks either internally or externally. They determine an end-state whose objective is to restore a built equilibrium through a large, often highly speculative single effort. They accomplish this by creating a large-scale node on existing networks. In order for such a plan to be attempted the state of disequilibrium in the built environment must have grown large enough to justify the immense expense of the new plan. This is why development will concentrate very large numbers of the same building program in one place, whether it is a cluster of 1000 identical single-family homes or a regional shopping mall, just like the skyscraper concentrates multiple identical floors in one place. Demand for these buildings has become so urgent that they can find a buyer despite the absence of local networks, the standardized building plan, or the monotonous setting. This is not as problematic for large cities for which a single subdivision is only a small share of the total urban fabric, but for smaller towns the same project can double the size of the urban fabric and overshoot the built equilibrium into an opposite and severe disequilibrium.

The mixed-used real estate development has attempted to recreate the sustainable features of the spontaneous city by imitating the morphology of sustainable local economic networks. It has not reintroduced the time dimension in economic network growth. Often this has resulted in a commercial sector that serves not the local neighborhood but the larger region first, consistent with the commercial sector being a product of large-scale economic network disequilibrium. In other developments the commercial sectors have struggled and been kept alive through subsidies from residential development, which is evidence of its unsustainability as part of the system.

In a medieval-era city the pace of urban growth is slow to a point where the growth of the city is not consciously noticed. Buildings are added sporadically, in random shape and order, as the extremely scarce economic situation makes no other pattern possible. Typically this means that the shape of streets will match the existing natural paths of movement, giving the street network an organic structure that is preserved through successive transformations in the urban fabric.

This works until the street network becomes large enough to become a functional problem. Because it is random, the medieval street network becomes complicated to move around in once the structure exceeds a certain scale. Some people see this as an obstacle to commerce and project to restructure the emergent medieval grid into something more rational. These projects fail for the same economic reasons that shaped the emergence of the medieval streets.

As the pace of urban growth increases and as the cartesian paradigm expands in the 17th and 18th centuries, deliberate city planning through the pre-emptive definition of an urban grid becomes fashionable. The practice of baroque planning remains the privilege of ultra-rich landlords considering the scale of construction involved. (Louis XIV’s Versailles is still the case study.) In the Americas such concentrations of capital do not yet exist. Grids are not truly part of a city plan, they are the outcome of regulations meant to avoid the pitfalls of medieval urban growth. Although the idea of a block is defined, the limiting shape of the grid itself is undefined. This allows cities to grow out, in theory, infinitely.

This works until the grid encounters and existing structure in the landscape. While Europe’s land is already very complex, in America the land is mostly empty. One exception is New York, which has multiple grids expanding towards the center of Manhattan, all with their own alignment with the waterfront. Compounding the medieval streets below Wall Street, the city’s network is getting messy. The solution conceived is the first city plan of New York, the Commissioners’ Plan of 1811, which grids Manhattan in the pattern it is famous for to this day with the help of a concentrated political power. In Europe this much centralization is not available, cities being ringed by a large number of villages that already structure the land. One notable exception is Barcelona, which under conservative military domination had reserved a large non aedificandi zone outside of its defensive walls. With the military out of the picture, and the industrial revolution putting enormous pressure on the city’s growth, the next most famous cartesian grid plan is imposed: the eixample. Adepts of the medieval city such as Camillo Sitte praise its artistic value and quality of life, but fail to truly describe how to reproduce it in the context of accelerating urbanization.

The 19th century is the triumph of the cartesian plan. It is not only employed to plan cities but to plan the entire American landscape. West of the original colonies the map becomes rectilinear. The flexibility and fluidity of New York’s grid plan promotes very rapid land development and the city achieves growth rates never before seen. European city planners are facing the same growth pressure but are trapped by the land’s existing structure, both physical and political. One simple solution is discovered: demolishing city walls and building a high capacity road that encircles the city, the boulevard. If it is to be complicated to get inside a city, it will at least be simple to get around it. Paris builds two on its two successive walls, and Vienna builds the famous Ringstrasse. An interesting phenomenon emerges from subsequent growth. While the boulevards were meant to be restful promenades, they emerge to become important centers on their own due to their attractiveness for traffic. In space syntax terms, they are integrators.

Manhattan’s grid extends to over a hundred streets but starts to suffer from severe scale problems. The medieval street system drives traffic away to boulevards, but in an endless grid traffic goes everywhere, and there is no place that is free of the increasing congestion. With the introduction of the car the endless grid is in crisis. Since no better idea is found, the grid system is replaced with the high-capacity collector road to concentrate all the congestion, from which huge, isolated developments access each other. This is the suburban sprawl system that remains the norm. It has the advantages of being simple to plan and giving enormous clout to land developers. However people are dissatisfied with the enormous scale of their environment. That they enjoy a single-family home does not sufficiently conceal the fact that they are clustered with thousands of similar homes, and next to those are huge strip malls, office parks and shopping malls that require long vehicle trips to access. The disconnect between their homes and their activities means they live in a form of crowded isolation. The suburbanites escaped congestion only to arrive at emptiness. There is more life in the less populated countryside. Adepts of the metropolitan grid such as Rem Koolhaas praise the culture of congestion as a lifestyle that the collector road fails to create.

This was as briefly stated as I could the modern history of the urban network: one system failing to adapt to the scale of the city, being replaced by a larger system that erases the small scale complexity of the previous only to itself fail at a much larger scale, and then another larger system crushing all complexity to resolve a problem of modernity.

Is there a way that we could have the benefits of all systems balanced as a whole urban network? To describe such a system, we can first define some proscriptions.

Any size of urban growth is allowed as long as the new growth extends the boundary of the network. This ensures that the city has the economic flexibility of the medieval city and allows anyone, no matter their economic importance, to contribute to the city’s growth.

The network must not become so complicated that it becomes impossible to move around in order to participate in large-scale activities and a culture of congestion.

Streets must not grow too long without interruption in such a way that speeding and traffic accidents are encouraged.

How does this work out in terms of prescriptions? It turns out to be very simple. If we assume that we start with a hamlet of a single block, or a regional road that is undeveloped, we need only two rules: one for private development and one for the community.

For private development: you may build on any available part of the network so long as you replace the part you used up by extending the network around your new block.

For community development: any time a part of the network becomes too complicated (for example it takes more than 4 steps to get out of a sector), extend the boundary of that part with a higher capacity road (a boulevard).

How do we tell if these two rules really do meet the proscriptions we defined? Since we’re talking about an emergent design, the only way to see how it works is to do an explicit simulation of the computations involved. For this I employed a Fibonacci sequence to stand for a random growth process. With each new block that the sequence generated, I placed it in the section of the network that minimized the private cost of extending the boundary. I also used square blocks to simplify the computations involved, and also to demonstrate how such a process would work in a structure of land that has been made square, for better of worse, through cartesian planning. The process would work just as well in a more fluid, rounder land structure such as exists in Europe and the American East.

Stage 1: The village

The village is a cluster of houses and small businesses, whose only real challenge is maintaining a facade with the outside by ensuring that every new block also fronts the countryside. This provides the village with a path that everyone can walk around on whenever they want to get some fresh air and open space.

Stage 2: The town

The town starts to support development at larger scales with bigger block sizes. The first boulevards are built around the original village, preserving its traditional atmosphere from the growing businesses on the new boulevards.

Stage 3: The city

Now a significant regional center, the city’s economic complexity is heralded by the construction of the ring road enclosing the town’s neighborhoods. Large developments such as a regional shopping mall, an airport and a TND line the ring road alongside other smaller blocks of more traditional housing and business that take advantage of the high centrality of the ring and its new culture of congestion, eventually forming whole neighborhoods of their own. The ring road also encloses available green spaces for recreation, making it a parkway in some segments.

Emergent properties of the process

The most interesting outcome is that the structure of the network makes a very nice chaotic fractal, showing the balance between scales in the city’s growth. It is simultaneously simple to grasp and complex, living geometry.

The spatial integration created by the boulevards and ring roads also promotes the creation of a hierarchy of different centers that are evenly distributed between neighborhoods. Tightly knit residential quarters provide security for children and the elderly, with neighborhood centers within walking distance and no threat of heavy traffic until the edge of the city, liberating citizens from automobile dependency.

Adopting a complex grid is going to benefit small towns and villages most, as their economy is typically not large enough to support the collector road system. It might even result in the emergence of new villages in rural regions that have experienced large-scale urbanization and thus make them more resilient to economic shocks.

For existing cities, history provides a precedent for increasing the grid’s complexity when the problem is scaling up the grid. The urban renovations of Haussmann in Paris or Robert Moses in New York showed how to compose a larger scale within an existing city. (In Moses’ case, how not to do so as well.) However there is no precedent for scaling down a network that is too big, which is what modern cities suffer from. I suspect that contrary to scaling up which requires a strong centralization of power, scaling down involves a decentralization and a multiplicity of new powers transforming neighborhoods, breaking up regional, municipal and even neighborhood authorities such as homeowners’ associations to create local economies.